78.21.5 problem 4 (c)

Internal problem ID [18370]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 31. Gauss Hypergeometric Equation. Problems at page 240
Problem number : 4 (c)
Date solved : Thursday, March 13, 2025 at 11:55:07 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (1-3 x \right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 31

Order:=6; 
ode:=x*(1-x)*diff(diff(y(x),x),x)+(1-3*x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (x^{5}+x^{4}+x^{3}+x^{2}+x +1\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 42

ode=x*(1-x)*D[y[x],{x,2}]+(1-3*x)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (x^5+x^4+x^3+x^2+x+1\right )+c_2 \left (x^5+x^4+x^3+x^2+x+1\right ) \log (x) \]

✓ Sympy. Time used: 0.823 (sec). Leaf size: 7

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x)*Derivative(y(x), (x, 2)) + (1 - 3*x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} + O\left (x^{6}\right ) \]